Knot polynomials for twist satellites

TL;DR

This paper develops a systematic approach to calculating knot polynomials for twist satellites, generalizing cabling, and explores their decomposition related to the $E_8$-sector and Vogel's universality, highlighting challenges in superpolynomial lifting.

Contribution

It introduces a general decomposition method for colored HOMFLY polynomials of twist satellites, extending previous cabling results and connecting to $E_8$-sector and Vogel's universality.

Findings

Decomposition of satellite's colored HOMFLY in original knot's polynomials.

Relation to $E_8$-sector and Vogel's universality.

Identified issues with lifting to superpolynomials for positive/negative twistings.

Abstract

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in those of the original knot, where contributing are adjoint and other representations from the "$E_8$-sector", what makes the story closely related to Vogel's universality. We also point out a problem with lifting the decomposition rule to the level of superpolynomials -- it looks like such rule, if any, should be different for positive and negative twistings.